Problem 21 - A Graphical Approach to College Algebra

Question

We can find an equation of a circle if we know the coordinates of the endpoints of a diameter of the circle. First, find the midpoint of the diameter, which is the center of the circle. Then find the radius, which is the distance from the center to either endpoint of the diameter. Finally use the center-radius form to find the equation. Find the center-radius form for each circle having the given endpoints of a diameter. $$(-1,3) \text { and }(5,-9)$$

Step-by-Step Solution

Answer

The circle's equation is \((x - 2)^2 + (y + 3)^2 = 45\).

Step 1: Find the Midpoint

The midpoint of the diameter is the center of the circle. To find the midpoint, use the midpoint formula: \[(x_m, y_m) = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\]Substitute the given points \((-1, 3)\) and \((5, -9)\): \[(x_m, y_m) = \left( \frac{-1 + 5}{2}, \frac{3 + (-9)}{2} \right) = (2, -3)\]The center of the circle is \((2, -3)\).

Step 2: Calculate the Radius

The radius is the distance from the center of the circle to any endpoint of the diameter. Use the distance formula:\[ r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]Using center \((2, -3)\) and endpoint \((-1, 3)\), we have:\[ r = \sqrt{(2 - (-1))^2 + (-3 - 3)^2} = \sqrt{(3)^2 + (-6)^2} = \sqrt{45} = 3\sqrt{5}\]The radius of the circle is \(3\sqrt{5}\).

Step 3: Write the Equation

Now, use the center-radius form of a circle's equation, which is:\[(x - h)^2 + (y - k)^2 = r^2\]Substitute \((h, k) = (2, -3)\) and \(r = 3\sqrt{5}\):\[(x - 2)^2 + (y + 3)^2 = (3\sqrt{5})^2 = 45\]Thus, the equation of the circle is \((x - 2)^2 + (y + 3)^2 = 45\).

Key Concepts

Midpoint Formula

The midpoint formula is a valuable tool in geometry for finding the center point between two endpoints on a line segment. It is particularly useful when working with diameters of circles to find the circle's center. In essence, the midpoint formula takes the average of the x-coordinates and the y-coordinates of the two endpoints.
For example, given two points \((-1, 3)\) and \(5, -9)\), the midpoint formula is:

• \((x_m, y_m) = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\)
• Substitute the points: \(x_m, y_m = \left( \frac{-1 + 5}{2}, \frac{3 + (-9)}{2} \right) = (2, -3)\)

Thus, the midpoint or center of the circle is \(2, -3)\).

Distance Formula

The distance formula is used to calculate the length of a line segment between two points in a coordinate plane. It is derived from the Pythagorean theorem and is instrumental in determining the radius of a circle. The distance between any two points \(x_1, y_1\) and \(x_2, y_2\) is given by:

• \(r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)

To find the radius of a circle when its diameter is given, the center and any endpoint can be used in this formula. For the given circle with the center \(2, -3)\) and an endpoint \((-1, 3)\), the radius \(r\) is calculated as:
\( r = \sqrt{(2 - (-1))^2 + (-3 - 3)^2} = \sqrt{45} = 3\sqrt{5} \)
This distance is essential in expressing the circle's equation in the center-radius form.

Center-Radius Form

The center-radius form of a circle's equation is a neat way to represent circles in a coordinate plane. It expresses the relationship involving the circle's center and radius. The form is:

• \((x - h)^2 + (y - k)^2 = r^2\)

where \(h, k\) represent the circle's center, and \(r\) is the radius. By substituting these into the formula, the equation becomes simple and standard. Given the center \(2, -3)\) and radius \(3\sqrt{5}\), we substitute these values into the center-radius form:
\( (x - 2)^2 + (y + 3)^2 = (3\sqrt{5})^2 = 45 \)
This equation beautifully encapsulates in a few terms the full essence of the circle regarding its position and size.

Geometry

Geometry is the branch of mathematics associated with shapes, sizes, and the properties of space. The study of circles is a fundamental part of geometry, involving concepts like area, diameter, and circumference.
When solving problems related to circles, understanding the spatial relationships in plane geometry, and using formulas such as the midpoint and distance formulas, becomes crucial.
Circles are described by their centers and radii, which are both aspects of geometry. Calculating the circle's equation in center-radius form provides a clear and efficient way to describe these shapes both in theoretical and applied geometry.